Introduction to Algorithms 3rd Edition Solutions

introduction to algorithms 3rd edition solutions

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Introduction: Introduction To Algorithms 3rd Edition Solutions

This part of Introduction To Algorithms 3rd Edition Solutions will start you thinking about designing and analyzing algorithms. It is intended to be a gentle introduction to how we specify algorithms, some of the design strategies we will use throughout this book(Introduction To Algorithms 3rd Edition Solutions), and many of the fundamental ideas used in algorithm analysis. Later parts of this book will build upon this base.
Chapter 1 provides an overview of algorithms and their place in modern computing systems. This chapter defines what an algorithm is and lists some examples. It also makes a case that we should consider algorithms as a technology, alongside technologies such as fast hardware, graphical user interfaces, object-oriented systems, and networks.

In Chapter 2, we see our first algorithms, which solve the problem of sorting a sequence of n numbers. They are written in a pseudocode which, although not directly translatable to any conventional programming language, conveys the structure of the algorithm clearly enough that you should be able to implement it in the language of your choice. The sorting algorithms we examine are insertion sort, which uses an incremental approach, and merges sort, which uses a recursive technique known as “divide-and-conquer.” Although the time each requires increases with the value of n, the rate of increase differs between the two algorithms. We determine these running times in Chapter 2, and we develop a useful notation to express them.

Introduction to Algorithms 3rd Edition Pdf Chapters and Sections

Table Of Contents For Introduction To Algorithms 3rd Edition Solutions


1 The Role of Algorithms in Computing

Algorithms as a technology

2 Getting Started

Insertion sort
Analyzing algorithms
Designing algorithms

3 Growth of Functions

Asymptotic notation
Standard notations and common functions

4 Divide-and-Conquer

The maximum-subarray problem
Strassen’s algorithm for matrix multiplication
The substitution method for solving recurrences
The recursion-tree method for solving recurrences
The master method for solving recurrences
Proof of the master theorem

5 Probabilistic Analysis and Randomized Algorithms

The hiring problem
Indicator random variables
Randomized algorithms
Probabilistic analysis and further uses of indicator random variables

II Sorting and Order Statistics

6 Heapsort

Maintaining the heap property
Building a heap
The heapsort algorithm
Priority Queues

7 Quicksort

Description of quicksort
Performance of quicksort
A randomized version of quicksort
Analysis of quicksort

8 Sorting in Linear Time

Lower bounds for sorting
Counting sort
Radix sort
Bucket sort

9 Medians and Order Statistics

Minimum and maximum
Selection in expected linear time
Selection in worst-case linear time

III Data Structures
10 Elementary Data Structures

Stacks and queues
Linked lists
Implementing pointers and objects
Representing rooted trees

11 Hash Tables

Direct-address tables
Hash tables
Hash functions
Open addressing
Perfect Hashing

12 Binary Search Trees

What is a binary search tree?
Querying a binary search tree
Insertion and deletion
Randomly built binary search trees

13 Red-Black Trees

Properties of red-black trees

14 Augmenting Data Structures

Dynamic order statistics
How to augment a data structure
Interval trees

IV Advanced Design and Analysis Techniques

15 Dynamic Programming

Rod cutting
Matrix-chain multiplication
Elements of dynamic programming
Longest common subsequence
Optimal binary search trees

16 Greedy Algorithms

An activity-selection problem
Elements of the greedy strategy
Huffman codes
Matroids and greedy methods
A task-scheduling problem as a matroid

17 Amortized Analysis

Aggregate analysis
The accounting method
The potential method
Dynamic tables

V Advanced Data Structures

Definition of B-trees
Basic operations on B-trees
Deleting a key from a B-tree

19 Fibonacci Heaps

Structure of Fibonacci heaps
Mergeable-heap operations
Decreasing a key and deleting a node
Bounding the maximum degree

20 van Emde Boas Trees

Preliminary approaches
A recursive structure
The van Emde Boas tree

21 Data Structures for Disjoint Sets

Disjoint-set operations
Linked-list representation of disjoint sets
Disjoint-set forests
Analysis of union by rank with path compression

VI Graph Algorithms

22 Elementary Graph Algorithms

Representations of graphs
Breadth-first search
Depth-first search
Topological sort
Strongly connected components

23 Minimum Spanning Trees

Growing a minimum spanning tree
The algorithms of Kruskal and Prim

24 Single-Source Shortest Paths

The Bellman-Ford algorithm
Single-source shortest paths in directed acyclic graphs
Dijkstra’s algorithm
Difference constraints and shortest paths
Proofs of shortest-paths properties

25 All-Pairs Shortest Paths

Shortest paths and matrix multiplication
The Floyd-Warshall algorithm
Johnson’s algorithm for sparse graphs

26 Maximum Flow

Flow networks
The Ford-Fulkerson method
Maximum bipartite matching
Push-relabel algorithms
The relabel-to-front algorithm

27 Multithreaded Algorithms

The basics of dynamic multithreading
Multithreaded matrix multiplication
Multithreaded merge sort

28 Matrix Operations

Solving systems of linear equations
Inverting matrices
Symmetric positive-definite matrices and least-squares approximation

29 Linear Programming

Standard and slack forms
Formulating problems as linear programs
The simplex algorithm
The initial basic feasible solution

30 Polynomials and the FFT

Representing polynomials
The DFT and FFT
Efficient FFT implementations

31 Number-Theoretic Algorithms

Elementary number-theoretic notions
Greatest common divisor
Modular arithmetic
Solving modular linear equations
The Chinese remainder theorem
Powers of an element
The RSA public-key cryptosystem
Primality testing
Integer factorization
String Matching
The naive string-matching algorithm
The Rabin-Karp algorithm
String matching with finite automata
The Knuth-Morris-Pratt algorithm
Computational Geometry
Line-segment properties
Determining whether any pair of segments intersects
Finding the convex hull
Finding the closest pair of points
Polynomial time
Polynomial-time verification
NP-completeness and reducibility
NP-completeness proofs
NP-complete problems
Approximation Algorithms
The vertex-cover problem
The traveling-salesman problem
The set-covering problem
Randomization and linear programming
The subset-sum problem

VIII Appendix: Mathematical Background

Summation formulas and properties
Bounding summations
Sets, Etc.
Counting and Probability
Discrete random variables
The geometric and binomial distributions
The tails of the binomial distribution
Matrices and matrix operations
Basic matrix properties

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